Properties

Label 1856.2.a.p.1.1
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +3.00000 q^{5} -2.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +3.00000 q^{5} -2.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} -3.00000 q^{13} +9.00000 q^{15} -4.00000 q^{17} +8.00000 q^{19} -6.00000 q^{21} +4.00000 q^{25} +9.00000 q^{27} +1.00000 q^{29} +3.00000 q^{31} +3.00000 q^{33} -6.00000 q^{35} +8.00000 q^{37} -9.00000 q^{39} -2.00000 q^{41} -7.00000 q^{43} +18.0000 q^{45} +11.0000 q^{47} -3.00000 q^{49} -12.0000 q^{51} -1.00000 q^{53} +3.00000 q^{55} +24.0000 q^{57} +4.00000 q^{59} -4.00000 q^{61} -12.0000 q^{63} -9.00000 q^{65} +4.00000 q^{67} -2.00000 q^{71} -12.0000 q^{73} +12.0000 q^{75} -2.00000 q^{77} -7.00000 q^{79} +9.00000 q^{81} -12.0000 q^{85} +3.00000 q^{87} -6.00000 q^{89} +6.00000 q^{91} +9.00000 q^{93} +24.0000 q^{95} -6.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 9.00000 2.32379
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −9.00000 −1.44115
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 18.0000 2.68328
\(46\) 0 0
\(47\) 11.0000 1.60451 0.802257 0.596978i \(-0.203632\pi\)
0.802257 + 0.596978i \(0.203632\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 24.0000 3.17888
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) −12.0000 −1.51186
\(64\) 0 0
\(65\) −9.00000 −1.11631
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 12.0000 1.38564
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 9.00000 0.933257
\(94\) 0 0
\(95\) 24.0000 2.46235
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) −18.0000 −1.75662
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −18.0000 −1.66410
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −21.0000 −1.84895
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −16.0000 −1.38738
\(134\) 0 0
\(135\) 27.0000 2.32379
\(136\) 0 0
\(137\) −20.0000 −1.70872 −0.854358 0.519685i \(-0.826049\pi\)
−0.854358 + 0.519685i \(0.826049\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 33.0000 2.77910
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) 0 0
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −24.0000 −1.94029
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −19.0000 −1.48819 −0.744097 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) 0 0
\(165\) 9.00000 0.700649
\(166\) 0 0
\(167\) −22.0000 −1.70241 −0.851206 0.524832i \(-0.824128\pi\)
−0.851206 + 0.524832i \(0.824128\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 48.0000 3.67065
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) 0 0
\(181\) 13.0000 0.966282 0.483141 0.875542i \(-0.339496\pi\)
0.483141 + 0.875542i \(0.339496\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) 24.0000 1.76452
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −18.0000 −1.30931
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) −27.0000 −1.93351
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 25.0000 1.72107 0.860535 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) −21.0000 −1.43219
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) −36.0000 −2.43265
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) 24.0000 1.60000
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) −25.0000 −1.63780 −0.818902 0.573933i \(-0.805417\pi\)
−0.818902 + 0.573933i \(0.805417\pi\)
\(234\) 0 0
\(235\) 33.0000 2.15268
\(236\) 0 0
\(237\) −21.0000 −1.36410
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.00000 0.441836 0.220918 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −36.0000 −2.25441
\(256\) 0 0
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −17.0000 −1.04826 −0.524132 0.851637i \(-0.675610\pi\)
−0.524132 + 0.851637i \(0.675610\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) −20.0000 −1.21942 −0.609711 0.792624i \(-0.708714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 0 0
\(273\) 18.0000 1.08941
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 72.0000 4.26491
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 0 0
\(293\) 34.0000 1.98630 0.993151 0.116841i \(-0.0372769\pi\)
0.993151 + 0.116841i \(0.0372769\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 14.0000 0.806947
\(302\) 0 0
\(303\) −24.0000 −1.37876
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) 29.0000 1.65512 0.827559 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) 0 0
\(309\) −18.0000 −1.02398
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 25.0000 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(314\) 0 0
\(315\) −36.0000 −2.02837
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) −3.00000 −0.165900
\(328\) 0 0
\(329\) −22.0000 −1.21290
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) 0 0
\(333\) 48.0000 2.63038
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 4.00000 0.217894 0.108947 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) 0 0
\(339\) 54.0000 2.93288
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) −27.0000 −1.44115
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 0 0
\(357\) 24.0000 1.27021
\(358\) 0 0
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) −30.0000 −1.57459
\(364\) 0 0
\(365\) −36.0000 −1.88433
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 1.00000 0.0517780 0.0258890 0.999665i \(-0.491758\pi\)
0.0258890 + 0.999665i \(0.491758\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 0 0
\(383\) −34.0000 −1.73732 −0.868659 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) 0 0
\(387\) −42.0000 −2.13498
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) −21.0000 −1.05662
\(396\) 0 0
\(397\) −19.0000 −0.953583 −0.476791 0.879017i \(-0.658200\pi\)
−0.476791 + 0.879017i \(0.658200\pi\)
\(398\) 0 0
\(399\) −48.0000 −2.40301
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) 0 0
\(405\) 27.0000 1.34164
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −60.0000 −2.95958
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 0 0
\(423\) 66.0000 3.20903
\(424\) 0 0
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) −9.00000 −0.434524
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 9.00000 0.431517
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) −9.00000 −0.425685
\(448\) 0 0
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) 30.0000 1.40952
\(454\) 0 0
\(455\) 18.0000 0.843853
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) −36.0000 −1.68034
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 27.0000 1.25210
\(466\) 0 0
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −66.0000 −3.04112
\(472\) 0 0
\(473\) −7.00000 −0.321860
\(474\) 0 0
\(475\) 32.0000 1.46826
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −11.0000 −0.502603 −0.251301 0.967909i \(-0.580859\pi\)
−0.251301 + 0.967909i \(0.580859\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −18.0000 −0.817338
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) −57.0000 −2.57763
\(490\) 0 0
\(491\) −5.00000 −0.225647 −0.112823 0.993615i \(-0.535989\pi\)
−0.112823 + 0.993615i \(0.535989\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 18.0000 0.809040
\(496\) 0 0
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) −66.0000 −2.94866
\(502\) 0 0
\(503\) −19.0000 −0.847168 −0.423584 0.905857i \(-0.639228\pi\)
−0.423584 + 0.905857i \(0.639228\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 0 0
\(513\) 72.0000 3.17888
\(514\) 0 0
\(515\) −18.0000 −0.793175
\(516\) 0 0
\(517\) 11.0000 0.483779
\(518\) 0 0
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) −24.0000 −1.04745
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 42.0000 1.81243
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 0 0
\(543\) 39.0000 1.67365
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 0 0
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 14.0000 0.595341
\(554\) 0 0
\(555\) 72.0000 3.05623
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 21.0000 0.888205
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) 54.0000 2.27180
\(566\) 0 0
\(567\) −18.0000 −0.755929
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −44.0000 −1.83174 −0.915872 0.401470i \(-0.868499\pi\)
−0.915872 + 0.401470i \(0.868499\pi\)
\(578\) 0 0
\(579\) 30.0000 1.24676
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.00000 −0.0414158
\(584\) 0 0
\(585\) −54.0000 −2.23263
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) −17.0000 −0.698106 −0.349053 0.937103i \(-0.613497\pi\)
−0.349053 + 0.937103i \(0.613497\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 0 0
\(597\) −6.00000 −0.245564
\(598\) 0 0
\(599\) 13.0000 0.531166 0.265583 0.964088i \(-0.414436\pi\)
0.265583 + 0.964088i \(0.414436\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) 0 0
\(605\) −30.0000 −1.21967
\(606\) 0 0
\(607\) 29.0000 1.17707 0.588537 0.808470i \(-0.299704\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −33.0000 −1.33504
\(612\) 0 0
\(613\) 27.0000 1.09052 0.545260 0.838267i \(-0.316431\pi\)
0.545260 + 0.838267i \(0.316431\pi\)
\(614\) 0 0
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) −31.0000 −1.24600 −0.622998 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 24.0000 0.958468
\(628\) 0 0
\(629\) −32.0000 −1.27592
\(630\) 0 0
\(631\) 22.0000 0.875806 0.437903 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(632\) 0 0
\(633\) 75.0000 2.98098
\(634\) 0 0
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) 9.00000 0.356593
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 0 0
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) −63.0000 −2.48062
\(646\) 0 0
\(647\) −20.0000 −0.786281 −0.393141 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) −18.0000 −0.705476
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) −36.0000 −1.40664
\(656\) 0 0
\(657\) −72.0000 −2.80899
\(658\) 0 0
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 0 0
\(663\) 36.0000 1.39812
\(664\) 0 0
\(665\) −48.0000 −1.86136
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 78.0000 3.01565
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) 36.0000 1.38564
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 54.0000 2.06928
\(682\) 0 0
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 0 0
\(685\) −60.0000 −2.29248
\(686\) 0 0
\(687\) −42.0000 −1.60240
\(688\) 0 0
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 0 0
\(693\) −12.0000 −0.455842
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) −75.0000 −2.83676
\(700\) 0 0
\(701\) −31.0000 −1.17085 −0.585427 0.810725i \(-0.699073\pi\)
−0.585427 + 0.810725i \(0.699073\pi\)
\(702\) 0 0
\(703\) 64.0000 2.41381
\(704\) 0 0
\(705\) 99.0000 3.72856
\(706\) 0 0
\(707\) 16.0000 0.601742
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) −42.0000 −1.57512
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −9.00000 −0.336581
\(716\) 0 0
\(717\) 60.0000 2.24074
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 51.0000 1.89671
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 28.0000 1.03562
\(732\) 0 0
\(733\) 48.0000 1.77292 0.886460 0.462805i \(-0.153157\pi\)
0.886460 + 0.462805i \(0.153157\pi\)
\(734\) 0 0
\(735\) −27.0000 −0.995910
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −1.00000 −0.0367856 −0.0183928 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) −72.0000 −2.64499
\(742\) 0 0
\(743\) −28.0000 −1.02722 −0.513610 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) 0 0
\(745\) −9.00000 −0.329734
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 21.0000 0.765283
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) −72.0000 −2.60317
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 48.0000 1.73092 0.865462 0.500974i \(-0.167025\pi\)
0.865462 + 0.500974i \(0.167025\pi\)
\(770\) 0 0
\(771\) 63.0000 2.26889
\(772\) 0 0
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) −48.0000 −1.72199
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −2.00000 −0.0715656
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −66.0000 −2.35564
\(786\) 0 0
\(787\) −46.0000 −1.63972 −0.819861 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) 0 0
\(789\) −51.0000 −1.81565
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) −9.00000 −0.319197
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) −44.0000 −1.55661
\(800\) 0 0
\(801\) −36.0000 −1.27200
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −60.0000 −2.11210
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) 0 0
\(813\) 39.0000 1.36779
\(814\) 0 0
\(815\) −57.0000 −1.99662
\(816\) 0 0
\(817\) −56.0000 −1.95919
\(818\) 0 0
\(819\) 36.0000 1.25794
\(820\) 0 0
\(821\) 37.0000 1.29131 0.645654 0.763630i \(-0.276585\pi\)
0.645654 + 0.763630i \(0.276585\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 0 0
\(829\) 28.0000 0.972480 0.486240 0.873825i \(-0.338368\pi\)
0.486240 + 0.873825i \(0.338368\pi\)
\(830\) 0 0
\(831\) −42.0000 −1.45696
\(832\) 0 0
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) −66.0000 −2.28402
\(836\) 0 0
\(837\) 27.0000 0.933257
\(838\) 0 0
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −39.0000 −1.34323
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 144.000 4.92470
\(856\) 0 0
\(857\) −11.0000 −0.375753 −0.187876 0.982193i \(-0.560160\pi\)
−0.187876 + 0.982193i \(0.560160\pi\)
\(858\) 0 0
\(859\) 51.0000 1.74010 0.870049 0.492966i \(-0.164087\pi\)
0.870049 + 0.492966i \(0.164087\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 42.0000 1.42804
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) −7.00000 −0.237459
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) −36.0000 −1.21842
\(874\) 0 0
\(875\) 6.00000 0.202837
\(876\) 0 0
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 0 0
\(879\) 102.000 3.44037
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 42.0000 1.41341 0.706706 0.707507i \(-0.250180\pi\)
0.706706 + 0.707507i \(0.250180\pi\)
\(884\) 0 0
\(885\) 36.0000 1.21013
\(886\) 0 0
\(887\) −1.00000 −0.0335767 −0.0167884 0.999859i \(-0.505344\pi\)
−0.0167884 + 0.999859i \(0.505344\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) 88.0000 2.94481
\(894\) 0 0
\(895\) 42.0000 1.40391
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 42.0000 1.39767
\(904\) 0 0
\(905\) 39.0000 1.29640
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) −48.0000 −1.59206
\(910\) 0 0
\(911\) −3.00000 −0.0993944 −0.0496972 0.998764i \(-0.515826\pi\)
−0.0496972 + 0.998764i \(0.515826\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −36.0000 −1.19012
\(916\) 0 0
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) −6.00000 −0.197922 −0.0989609 0.995091i \(-0.531552\pi\)
−0.0989609 + 0.995091i \(0.531552\pi\)
\(920\) 0 0
\(921\) 87.0000 2.86675
\(922\) 0 0
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) −36.0000 −1.18240
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 75.0000 2.44753
\(940\) 0 0
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −54.0000 −1.75662
\(946\) 0 0
\(947\) −51.0000 −1.65728 −0.828639 0.559784i \(-0.810884\pi\)
−0.828639 + 0.559784i \(0.810884\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15.0000 0.485898 0.242949 0.970039i \(-0.421885\pi\)
0.242949 + 0.970039i \(0.421885\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.00000 0.0969762
\(958\) 0 0
\(959\) 40.0000 1.29167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 30.0000 0.965734
\(966\) 0 0
\(967\) 59.0000 1.89731 0.948656 0.316310i \(-0.102444\pi\)
0.948656 + 0.316310i \(0.102444\pi\)
\(968\) 0 0
\(969\) −96.0000 −3.08396
\(970\) 0 0
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −36.0000 −1.15292
\(976\) 0 0
\(977\) 37.0000 1.18373 0.591867 0.806035i \(-0.298391\pi\)
0.591867 + 0.806035i \(0.298391\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) −66.0000 −2.10080
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 0 0
\(993\) −9.00000 −0.285606
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) −16.0000 −0.506725 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(998\) 0 0
\(999\) 72.0000 2.27798
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1856.2.a.p.1.1 1
4.3 odd 2 1856.2.a.b.1.1 1
8.3 odd 2 464.2.a.f.1.1 1
8.5 even 2 58.2.a.a.1.1 1
24.5 odd 2 522.2.a.k.1.1 1
24.11 even 2 4176.2.a.bh.1.1 1
40.13 odd 4 1450.2.b.f.349.2 2
40.29 even 2 1450.2.a.i.1.1 1
40.37 odd 4 1450.2.b.f.349.1 2
56.13 odd 2 2842.2.a.d.1.1 1
88.21 odd 2 7018.2.a.c.1.1 1
104.77 even 2 9802.2.a.d.1.1 1
232.133 odd 4 1682.2.b.e.1681.2 2
232.157 odd 4 1682.2.b.e.1681.1 2
232.173 even 2 1682.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.a.1.1 1 8.5 even 2
464.2.a.f.1.1 1 8.3 odd 2
522.2.a.k.1.1 1 24.5 odd 2
1450.2.a.i.1.1 1 40.29 even 2
1450.2.b.f.349.1 2 40.37 odd 4
1450.2.b.f.349.2 2 40.13 odd 4
1682.2.a.j.1.1 1 232.173 even 2
1682.2.b.e.1681.1 2 232.157 odd 4
1682.2.b.e.1681.2 2 232.133 odd 4
1856.2.a.b.1.1 1 4.3 odd 2
1856.2.a.p.1.1 1 1.1 even 1 trivial
2842.2.a.d.1.1 1 56.13 odd 2
4176.2.a.bh.1.1 1 24.11 even 2
7018.2.a.c.1.1 1 88.21 odd 2
9802.2.a.d.1.1 1 104.77 even 2